ME 461:
Finite Element
Analysis
Fall
2015
A Semester Report on:
The Development and Analysis of a Gear Train
Group Members:
Brandon Drust, Nick Kirkland, Matt Krotowski, Matt Vincent
Department of Mechanical and Nuclear Engineering, University Park, PA
The Development and Analysis of a Gear Train
Table of Contents
Table of Contents .................................................................................................................. 2
Executive Summary ............................................................................................................... 3
Acknowledgements …………………………………………………………………………………………………………….. 4
List of Figures and Tables ....................................................................................................... 5
Section 1: Background and Project Plan ................................................................................. 6
Section 2: Development and Description of the CAD Geometry .............................................. 7
Section 3.1: Development of Finite Element Meshes in Abaqus .............................................. 9
Section 3.2: Development of Finite Element Meshes in Nastran ........................................... 10
Section 4.1: Development and Description of the Model Assembly and Boundary Conditions
in Abaqus ............................................................................................................................ 12
Section 4.2: Development and Description of the Model Assembly and Boundary Conditions
in Nastran ........................................................................................................................... 14
Section 5.1: Development and Description of Model Interactions in Abaqus ........................ 15
Section 5.2: Development and Description of Model Interactions in Nastran ........................ 17
Section 6: Analysis of Finite Element Model ......................................................................... 18
Section 7: Summary of Major Findings ................................................................................. 21
Section 8: Works Cited......................................................................................................... 22
2
The Development and Analysis of a Gear Train
Executive Summary
This analytical report shows the stress and strain components formed from gears rotating in a
gear train. There are five gears present in this gear train: a 14 tooth gear, a 24 tooth gear, a 30
tooth gear, a 40 tooth gear, and a 48 tooth gear. These gears all contain a 20° pressure angle and
a pitch of 48. The 14 tooth gear is the driver gear and it is rotating with a torque of 21.4 N-m.
Each gear experiences a contact friction coefficient of 0.15 between itself and the shaft, which is
a common friction coefficient for rods rotating on bearings.
Models of the five gears and the base were created in Solidworks while the finite element
analysis (FEA) was done primarily in Nastran. The gears were meshed with a global size of 0.01.
These sizes provided a small enough mesh around the teeth in order to provide an analysis of
stress and strain.
In order to limit each gear rotationally, each gear was locked rotationally in the x direction
around its shaft. Finally a clockwise torque was placed on the 14 tooth gear. This torque would
cause the teeth to contact the teeth on the 24 tooth gear which will cause the rest of the gear train
to rotate as well.
Results were obtained from both Abaqus and Nastran. The Abaqus results showed that the
maximum stress was along the inner surface of the gear where it contacts the rod. The Nastran
results showed that the maximum stress was located in the base of the contact tooth in the
driving gear (14 tooth) and were found to be .0226 MPA. This shows that the gear train will not
yield under this application. Nastran also showed that the maximum rotational displacement in
each gear. The driving gear had the most rotational displacement and the last gear in the gear
train had the least amount of rotational displacement.
3
The Development and Analysis of a Gear Train
Acknowledgements
The gear train team would like to thank both Dr. Reuben Kraft and Dooman Akbarian for their
help in achieving valid results. They were critical assets in determining what would work in
Abaqus versus what could not work in Abaqus.
The gear train team would also like to thank McMaster Carr for providing the team with the
dimensions of the gears to run the analysis with.
The gear train team would like to thank GrabCAD for the Solidworks Models of the gears, so the
team could import those files into the appropriate Finite Element Modeling softwares.
The gear train team would also like to thank Nastran for being able to run the job that we
submitted and provide the team with valid results.
4
The Development and Analysis of a Gear Train
List of Figures and Tables
Figure 1………………………………………………………………………………………………………………….…………….. 6
Figure2.1……………………………………………………………………………………………………………………….…….. 7
Figure2.2…………………………………………………………………………………………………………………….……….. 8
Figure 2.3……………………………………………………………………………………………………………………………… 8
Figure 2.4……………………………………………………………………………………………………………………………… 8
Figure2.5…………………………………………………………………………………..…………………………………………. 8
Figure2.6…………………………………………………..…………………………………………………………………………. 8
Figure 2.7………………………………………………………………………………………………................................ 8
Table 2.1………………………………………………………………………………………………………………………………. 8
Table 3.1………………………………………………………………………………………………………………………………. 9
Table3.2………………………………………………………………………………………………………………………..…….. 9
Figure3.1…………………………………………………………..…………………………………………………………………. 9
Figure3.2………………………………………………….……………………………………………………………………….. 10
Figure4.1………………………………………………………..…………………………………………………………………. 10
Figure4.2……………………………………………………………………………………………………………………………. 11
Figure 5.1……………………….………………………………………………………………………………………………….. 12
Figure5.2…………………………….……………………………………………………………………………………………… 12
5
The Development and Analysis of a Gear Train
Section 1: Background and Project Plan
One of the most fundamental components of mechanical engineering is a spur gear.
Understanding the stress, strain, location, and magnitude of these components allows for more
efficient designs of gears and gear systems. Many times, gears are oriented into a gear train,
which is when gears are mounted to a frame so the teeth engage. Each time a tooth engages, a
force is placed on the gear, which means it experiences a stress. For this project, a gear train
containing several varying sized gears will be analyzed to determine how the size, shape, and
rotational speed affects each individual gear.
Each team member will be responsible for designing a gear with a different size and number of
teeth. Before starting, the team will determine the designated pressure angle and pitch to make
sure the gears will be compatible. The gears will then be oriented into a gear train similar to
Figure 1 with one gear being the driver. That gear will rotate at a specified torque, turning the
other gears. Stresses on the teeth of each gear will then be determined using Abaqus and Nastran
to do the finite element analysis.
Figure : Gear Train
Figure 1: Simple Configuration of a Gear Train with Driver, Idler, and Follower
6
The Development and Analysis of a Gear Train
Section 2: Development and Description of the CAD Geometry
Figure 2.1: Labeled Diagram of Pinion and Gear Assembly [1]
The various parts of the gear train were designed in Solidworks from a gear template found
online. Five different gears were created with sizes of 14 Teeth, 24 Teeth, 30 Teeth, 40 Teeth,
and 48 Teeth. These models all include a 20° pressure angle and a pitch of 48 to be compatible
with one another. Figures 2.2-2.7 represent the gears that will be assembled into the gear train
and the base that holds them.
7
The Development and Analysis of a Gear Train
Figure 2.2: Gear 1 (14 Tooth Gear)
Figure 2.3: Gear 2 (24 Tooth Gear)
Figure 2.4: Gear 3 (30 Tooth Gear)
Figure 2.5: Gear 4 (40 Tooth Gear)
Figure 2.6: Gear 5 (48 Tooth Gear)
Figure 2.7: Gear Train Base
In order to properly model each gear, the units were first set to metric to make sure that the
Abaqus and Nastran analysis would be compatible. To alter the size of each gear, the number of
required teeth was inserted into the parametric equations of the gear template. The gear equation
solver in Solidworks then calculated the associated dimensions of each gear. These dimensions
can be found in Table 2.1 below.
Table 2.1: Dimensions of All Gears in Gear Train
Gear
# of
Teeth
Dia.
Pitch
(mm)
1
2
3
4
5
14
24
30
40
48
203.2
203.2
203.2
203.2
203.2
Pressure Addendum Dedendum Pitch Base Clearance Width
Angle
(mm)
(mm)
Circle Circle
(Deg.)
Dia.
Dia.
20
20
20
20
20
3.175
3.175
3.175
3.175
3.175
3.969
3.969
3.969
3.969
3.969
44.45
76.2
95.25
127
152.4
41.65
71.60
89.51
119.3
143.2
0.794
0.794
0.794
0.794
0.794
25.4
25.4
25.4
25.4
25.4
The final step in the development of the CAD geometry was saving the gears as STEP files to be
able to successfully import the parts into both the Abaqus and Nastran programs.
8
The Development and Analysis of a Gear Train
Section 3.1: Development of Finite Element Meshes in Abaqus
To develop the meshes for the gears, the STEP files of each model were imported to Abaqus and
Nastran. A material was then created and assigned to each part of the gear train. This material
was given the properties of steel which can be seen in table 3.1 below.
Table 3.1: Properties of the Material Steel Assigned to each Gear
Gear
Elastic Modulus
(GPa)
200
All
Poisons Ratio
Density (kg/m3)
0.3
7800
Table 3.2: Mass of Gears and Base
Part
Mass (kg)
Gear 1
0.225
Gear 2
0.808
Gear 3
1.26
Gear 4
2.35
Gear 5
3.44
Base
26.39
In addition to creating and assigning a material, a step was created named Step-1. Step-1 was a
static, general type with an initial incrimination set to 0.1.
After creating both a material and a step, each gear was meshed individually. This was done by
assigning a mesh control and element type in the individual part view. For both the mesh control
and element type, a Tet mesh and default element type were selected. The base was meshed with
a global size of 0.05 and the gear with a global size of 0.002. The completed mesh can be seen in
Figure 3.1 and a close-up of the teeth in Figure 3.2.
Figure 3.1: Completed mesh with Global Size of 0.002 for each Gear
9
The Development and Analysis of a Gear Train
Figure 3.2: Close-up of Mesh between Gear Teeth
Section 3.2: Development of Finite Element Meshes in Nastran
To develop meshes in Nastran, an assembly was first created in NX as shown in Figure 3.3. To
do this, the gears were placed with their centers zeroed in the y direction, and they were moved
horizontally by the distance of their pitch diameter to ensure the teeth will interface correctly.
Next, pins were modeled to fit into the key of the gear. They were then placed in the assembly so
they were flush with the front face of each gear.
Figure 3.3: Gear Train in NX
After the assembly was created, it was ported over to Nastran where each part was meshed
independently with tet10 elements, a global mesh size of 0.01 m, and the material property of
steel as shown in Figure 3.4. The completed mesh can be seen in Figure 3.5. This mesh had to be
created rather large in order to cut down on run time.
10
The Development and Analysis of a Gear Train
Figure 3.4: Nastran Mesh Properties and Mesh of Teeth
Figure 3.5. Completed Mesh in Nastran
11
The Development and Analysis of a Gear Train
Section 4.1: Development and Description of the Model Assembly and
Boundary Conditions in Abaqus
After assigning the material and creating a mesh, the gears needed to be oriented in a position
with which they could interact with each other. In order to do this, each gear was assembled onto
the base. Each gear was translated to its appropriate peg on the base in order to keep the
distances between gears fixed. One side of each gear was aligned with the end of its respective
peg to ensure that all the gears were aligned. They were then rotated slightly to create a perfect
fit that can be seen in Figure 4.1 below.
Figure 4.1: Close-up of Gear Teeth
To keep the base stationary, a displacement/rotation boundary condition was applied to the back
surface of the base. All directions of the boundary condition were set to 0 to ensure that the base
would not move during the analysis. This boundary condition is shown in Figure 4.2.
12
The Development and Analysis of a Gear Train
Figure 4.2: Initial Boundary Condition for Base Plate of Assembly
A load was then applied to the driver/pinion gear (gear 1). This load was applied using a moment
acting on the edge of one of the teeth. A random tooth edge was chosen because the moment acts
on the entire gear, which means the location of its application is arbitrary. The magnitude of the
moment was chosen to be 21.4 N-m. This magnitude was found to be the average torque for a
small electric motor.
13
The Development and Analysis of a Gear Train
Section 4.2: Development and Description of the Model Assembly and
Boundary Conditions in Nastran
After assigning a mesh and material in Nastran, boundary conditions and loads needed to be
established in order to get the solution to run. Since the pinon gear is the only gear driving this
assembly, only one load was applied in the analysis. A torque of 21.4 N-m in the clockwise
direction was placed on the surface of pin in order to simulate an electric motor applying this
load. This can be seen in Figure 4.3.
Figure 4.3: Torque Applied to Pinon Gear Pin
Boundary conditions were then applied to the assembly in order to restrict movement in certain
directions to allow for accurate results. A pin joint was attached to the center of the pins to alow
the pins to rotate freely but not displace in any direction. A friction factor of 0.15 was added to
this boundary condition to simulate the pins rotating on greased roller bearings. This can be seen
in Figure 4.4.
14
The Development and Analysis of a Gear Train
Figure 4.4. Assembly Fixed on Pin Constraints.
Section 5.1: Development and Description of Model Interactions in
Abaqus
The first interaction applied to the gear train was between each of the gears and the shaft that
each of them rotate on. A contact property with tangential behavior and a coefficient of friction
of 0.15 was applied to each gear and rod pair. The coefficient of friction of 0.15 was chosen
because it is a common friction coefficient for rods on bearings. The rod was signified as the
master surface and the gear as the slave surface. Figures 5.1 and 5.2 show contact property
details and the surfaces to which it was applied.
15
The Development and Analysis of a Gear Train
Figure 5.1: Contact Properties for Gears on Base Plate
Figure 5.2: Master and Slave Surface For Fricional Contact Property
16
The Development and Analysis of a Gear Train
Section 5.2: Development and Description of Model Interactions in
Nastran
Two model interactions were placed in Nastran. First, contacts between the pin and gear were
created to allow the pin to rotate the gear when the load was applied. This can be seen in Figure
5.3. This interaction simulated the gear being pressfit onto the shaft and roatating with the pin.
Figure 5.3: Contact Interaction between Pin and Gear
Second, contact interactions were placed between meshing teeth of the gears with a friction
coefficient of 0.74 to represent steel on steel sliding. This interaction allowed the teeth to make
contact with eachother causing the next gear in line to rotate and experience stress from the
loading. This can be seen in Figure 5.4.
Figure 5.4: Contact Interaction between Teeth on Meshing Gears
17
The Development and Analysis of a Gear Train
Section 6: Analysis of Finite Element Model
The gear train team was not able to run the analysis within Abaqus. There were several attempts
made with different boundary conditions, interactions, etc. to try and mitigate the error messages
that kept coming up for each job submission. In order to resolve this issue, the finite element
models were also developed in Nastran, where the team was able to run the job and observe the
stresses and strain on the gears in the gear train.
Figure 6.1: Stress Interactions between all Gear Sets
The results from the job conclude that the stress will dissipate from the first to the last gear. The
stress seems to propagate radially inward from each contact surface. The stress concentrations
will be more concentrated in the first gear because the initial gear is taking all of the stress of the
inertia from starting to spin the other gears throughout the gear train. The inertia is present
because of the sliding friction being created from the roller bearings located on the gear
attachment in the center of each gear.
18
The Development and Analysis of a Gear Train
Figure 6.3: Stress Concentration Between Gear 1 and 2
Figure 6.4: Stress Concentration Between Gear 2 and 3
Figure 6.6: Stress Concentration Between Gear 4 and 5
Figure 6.5: Stress Concentration Between Gear 3 and 4
The figures above outline the stress concentrations developed during the eighth iteration of the
job ran in Nastran. The stress concentrations are the highest between the teeth on the first and
second gear, which was anticipated. The involute curves on each gear put an excessive amount
of stress in the form of point loads on each gear tooth. The gear teeth above show the point loads
that are exhibited during the job, especially in Figure 6.3 where the base of the tooth of gear one
is exhibiting high amounts of stress from the tooth on gear two.
19
The Development and Analysis of a Gear Train
Figure 6.2: Localized Strain Displacements for each Gear
The results in Figure 6.2 above exhibit a displacement of each of the gear teeth involved in the
eighth iteration. The gear teeth in gear one exhibit the highest displacement (once again because
of the high torque created from the applied torque and the friction of the other gears). The
displacement then dissipates radially inward for each of the subsequent gears (similar to the
stress concentrations).
20
The Development and Analysis of a Gear Train
Section 7: Summary of Major Findings
Obtaining results for the entire gear train in Abaqus resulted in a job that would have taken
multiple hours to complete with the computer power available. The same analysis in Nastran
took considerably less time and computing power to produce the same results. The boundary
conditions were also much easier to apply in Nastran, making for an all-around easier analysis.
The rotating nature of this analysis played a major role in the difficulty of the project. The
rotation forced the group to look at different boundary conditions that do not be considered in a
simple linear translational analysis.
The max stress found in the gear train was 0.0226 MPa. Comparing this to the yield strength of
steel (250 MPa), the group concluded that the gear teeth would not break under the torque
applied. The gear train under investigation would actually be able to handle a torque from a
much more powerful motor.
21
The Development and Analysis of a Gear Train
Section 8: Works Cited
"AutomationDirect | The Common Sense Way to Buy Industrial Controls." AutomationDirect |
The Common Sense Way to Buy Industrial Controls. N.p., n.d. Web. 20 Oct. 2015.
"Bostongear.com Your Source for Speed Reducers, Gears and Power Transmission Components
for Industrial Applications." Bostongear.com Your Source for Speed Reducers, Gears and
Power Transmission Components for Industrial Applications. N.p., n.d. Web. 5 Oct.
2015.
"Friction and Coefficients of Friction." Friction and Coefficients of Friction. N.p., n.d. Web. 10
Oct. 2015.
"GrabCAD - CAD Library." GrabCAD - CAD Library. N.p., n.d. Web. 26 Sept. 2015.
22